# ADMM

### *class* deepinv.optim.ADMM(data_fidelity=None, prior=None, lambda_reg=1.0, stepsize=1.0, beta=1.0, g_param=None, sigma_denoiser=None, max_iter=100, crit_conv='residual', thres_conv=1e-5, early_stop=False, custom_metrics=None, custom_init=None, unfold=False, trainable_params=None, g_first=False, cost_fn=None, params_algo=None, \*\*kwargs)

Bases: [`BaseOptim`](https://deepinv.org/api/stubs/deepinv.optim.BaseOptim.html.md#deepinv.optim.BaseOptim)

ADMM module for solving the problem

$$
\begin{equation}
\label{eq:min_prob}
\tag{1}
\underset{x}{\arg\min} \quad  \datafid{x}{y} + \lambda \reg{x},
\end{equation}

$$

where $\datafid{x}{y}$ is the data-fidelity term, $\reg{x}$ is the regularization term.
If the attribute `g_first` is set to False (by default), the ADMM iterations write (see Boyd *et al.*<sup>[1](#footcite-boyd2011distributed)</sup> for more details):

$$
u_{k+1} &= \operatorname{prox}_{\gamma f}(x_k - z_k) \\
x_{k+1} &= \operatorname{prox}_{\gamma \lambda \regname}(u_{k+1} + z_k) \\
z_{k+1} &= z_k + \beta (u_{k+1} - x_{k+1})

$$

where $\gamma>0$ is a stepsize and $\beta>0$ is a relaxation parameter.  If the attribute `g_first` is set to `True`, the functions $f$ and $\regname$ are
inverted in the previous iterations. The ADMM iterations are defined in the iterator class [`deepinv.optim.optim_iterators.ADMMIteration`](https://deepinv.org/api/stubs/deepinv.optim.optim_iterators.ADMMIteration.html.md#deepinv.optim.optim_iterators.ADMMIteration).
For using early stopping or stepsize backtracking, see the documentation of the [`deepinv.optim.BaseOptim`](https://deepinv.org/api/stubs/deepinv.optim.BaseOptim.html.md#deepinv.optim.BaseOptim) class.

If the attribute `unfold` is set to `True`, the algorithm is unfolded and the algorithmic parameters (stepsize, regularization parameter, etc.) of the algorithm are trainable.
By default (if the attribute `unfold` is set to `True`) all the algorithm parameters are trainable: the stepsize $\gamma$, the regularization parameter $\lambda$, the prior parameter and the relaxation parameter $\beta$.
Use the `trainable_params` argument to adjust the list of trainable parameters.
Note also that by default, if the prior has trainable parameters (e.g. a neural network denoiser), these parameters are learnable by default.
If the model is used for inference only, use the `with torch.no_grad():` context when calling the model in order to avoid unnecessary gradient computations.

* **Parameters:**
  * **data_fidelity** ([*list*](https://docs.python.org/3.12/library/stdtypes.html#list) *,* [*deepinv.optim.DataFidelity*](https://deepinv.org/api/stubs/deepinv.optim.DataFidelity.html.md#deepinv.optim.DataFidelity)) – data-fidelity term $\datafid{x}{y}$.
    Either a single instance (same data-fidelity for each iteration) or a list of instances of
    [`deepinv.optim.DataFidelity`](https://deepinv.org/api/stubs/deepinv.optim.DataFidelity.html.md#deepinv.optim.DataFidelity) (distinct data fidelity for each iteration). Default: `None` corresponding to $\datafid{x}{y} = 0$.
  * **prior** ([*list*](https://docs.python.org/3.12/library/stdtypes.html#list) *,* [*deepinv.optim.Prior*](https://deepinv.org/api/stubs/deepinv.optim.Prior.html.md#deepinv.optim.Prior)) – regularization prior $\reg{x}$.
    Either a single instance (same prior for each iteration) or a list of instances of
    [`deepinv.optim.Prior`](https://deepinv.org/api/stubs/deepinv.optim.Prior.html.md#deepinv.optim.Prior) (distinct prior for each iteration). Default: `None` corresponding to $\reg{x} = 0$.
  * **lambda_reg** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – regularization parameter $\lambda$. Default: `1.0`.
  * **stepsize** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – stepsize parameter $\gamma$. Default: `1.0`.
  * **beta** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – ADMM relaxation parameter $\beta$. Default: `1.0`.
  * **g_param** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – parameter of the prior function. For example the noise level for a denoising prior. Default: `None`.
  * **sigma_denoiser** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – same as `g_param`. If both `g_param` and `sigma_denoiser` are provided, `g_param` is used. Default: `None`.
  * **max_iter** ([*int*](https://docs.python.org/3.12/library/functions.html#int)) – maximum number of iterations of the optimization algorithm. Default: `100`.
  * **crit_conv** ([*str*](https://docs.python.org/3.12/library/stdtypes.html#str)) – convergence criterion to be used for claiming convergence, either `"residual"` (residual
    of the iterate norm) or `"cost"` (on the cost function). Default: `"residual"`
  * **thres_conv** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – convergence threshold for the chosen convergence criterion. Default: `1e-5`.
  * **early_stop** ([*bool*](https://docs.python.org/3.12/library/functions.html#bool)) – whether to stop the algorithm as soon as the convergence criterion is met. Default: `False`.
  * **custom_metrics** ([*dict*](https://docs.python.org/3.12/library/stdtypes.html#dict)) – dictionary of custom metric functions to be computed along the iterations. The keys of the dictionary are the names of the metrics, and the values are functions that take as input the current and previous iterates, and return a scalar value. Default: `None`.
  * **custom_init** (*Callable*) – 

    Custom initialization of the algorithm.
    The callable function `custom_init(y, physics)` takes as input the measurement $y$ and the physics `physics` and returns the initialization in the form of either:
    - a tuple $(x_0, z_0)$ (where `x_0` and `z_0` are the initial primal and dual variables),
    - a torch.Tensor $x_0$ (if no dual variables $z_0$ are used), or
    - a dictionary of the form `X = {'est': (x_0, z_0)}`.

    Note that custom initialization can also be directly defined via the `init` argument in the `forward` method.

    If `None` (default value), the algorithm is initialized with the adjoint $A^{\top}y$ when the adjoint is defined,
    and with the observation `y` if the adjoint is not defined. Default: `None`.
  * **g_first** ([*bool*](https://docs.python.org/3.12/library/functions.html#bool)) – whether to perform the proximal step on $\reg{x}$ before that on $\datafid{x}{y}$, or the opposite. Default: `False`.
  * **unfold** ([*bool*](https://docs.python.org/3.12/library/functions.html#bool)) – whether to unfold the algorithm or not. Default: `False`.
  * **trainable_params** ([*list*](https://docs.python.org/3.12/library/stdtypes.html#list)) – list of ADMM parameters to be trained if `unfold` is True. To choose between `["lambda", "stepsize", "g_param", "beta"]`. Default: None, which means that all parameters are trainable if `unfold` is True. For no trainable parameters, set to an empty list.
  * **cost_fn** (*Callable*) – Custom user input cost function.
    `cost_fn(x, data_fidelity, prior, cur_params, y, physics)` takes as input
    the current primal variable ([`torch.Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)), the current data-fidelity ([`deepinv.optim.DataFidelity`](https://deepinv.org/api/stubs/deepinv.optim.DataFidelity.html.md#deepinv.optim.DataFidelity)),
    the current prior ([`deepinv.optim.Prior`](https://deepinv.org/api/stubs/deepinv.optim.Prior.html.md#deepinv.optim.Prior)), the current parameters (dict), and the measurement ([`torch.Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)).
    Default: `None`.
  * **params_algo** ([*dict*](https://docs.python.org/3.12/library/stdtypes.html#dict)) – optionally, directly provide the ADMM parameters in a dictionary. This will overwrite the parameters in the arguments `stepsize`, `lambda_reg`, `g_param` and `beta`.

<hr />

* **References:**

* <a id='footcite-boyd2011distributed'>**[1]**</a> Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, Jonathan Eckstein, and others. Distributed optimization and statistical learning via the alternating direction method of multipliers. *Foundations and Trends® in Machine learning*, 3(1):1–122, 2011.

<a id="sphx-glr-backref-deepinv-optim-admm"></a>

## Examples using `ADMM`:

<div class="sphx-glr-thumbnails">
<!-- thumbnail-parent-div-open --><div class="sphx-glr-thumbcontainer" tooltip="In this example, we show how to solve a deblurring inverse problem using an explicit prior.">![](auto_examples/optimization/images/thumb/sphx_glr_demo_custom_prior_thumb.png)

[Image deblurring with custom deep explicit prior.](https://deepinv.org/auto_examples/optimization/demo_custom_prior.html.md)

  <div class="sphx-glr-thumbnail-title">Image deblurring with custom deep explicit prior.</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This demo shows the use of the deepinv.physics.SpatialUnwrapping forward model and the deepinv.optim.ItohFidelity for unwrapping problems, which occur in modulo imaging, interferometry SAR and other imaging applications. It shows how to generate a wrapped phase image, apply blur and noise, and reconstruct the original phase using both DCT inversion and ADMM optimization.">![](auto_examples/physics/images/thumb/sphx_glr_demo_spatial_unwrapping_thumb.png)

[Spatial unwrapping and modulo imaging](https://deepinv.org/auto_examples/physics/demo_spatial_unwrapping.html.md)

  <div class="sphx-glr-thumbnail-title">Spatial unwrapping and modulo imaging</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This demo illustrates the impact of different Hadamard pattern ordering algorithms in the Single Pixel Camera (SPC), a computational imaging system that uses a single photodetector to capture images by projecting the scene onto a series of patterns. The SPC is implemented in the DeepInverse library.">![](auto_examples/physics/images/thumb/sphx_glr_demo_spc_thumb.png)

[Pattern Ordering in a Compressive Single Pixel Camera](https://deepinv.org/auto_examples/physics/demo_spc.html.md)

  <div class="sphx-glr-thumbnail-title">Pattern Ordering in a Compressive Single Pixel Camera</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example shows how to define your own optimization algorithm. For example, here, we implement the Primal-Dual Condat-Vu (CV) algorithm, and apply it for Single Pixel Camera reconstruction.">![](auto_examples/plug-and-play/images/thumb/sphx_glr_demo_PnP_custom_optim_thumb.png)

[PnP with custom optimization algorithm (Primal-Dual Condat-Vu)](https://deepinv.org/auto_examples/plug-and-play/demo_PnP_custom_optim.html.md)

  <div class="sphx-glr-thumbnail-title">PnP with custom optimization algorithm (Primal-Dual Condat-Vu)</div>
</div>
<!-- thumbnail-parent-div-close --></div>